584 research outputs found

    Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems

    Get PDF
    The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles

    Persistence of instanton connections in chemical reactions with time dependent rates

    Get PDF
    The evolution of a system of chemical reactions can be studied, in the eikonal approximation, by means of a Hamiltonian dynamical system. The fixed points of this dynamical system represent the different states in which the chemical system can be found, and the connections among them represent instantons or optimal paths linking these states. We study the relation between the phase portrait of the Hamiltonian system representing a set of chemical reactions with constant rates and the corresponding system when these rates vary in time. We show that the topology of the phase space is robust for small time-dependent perturbations in concrete examples and state general results when possible. This robustness allows us to apply some of the conclusions on the qualitative behavior of the autonomous system to the time-dependent situation

    Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

    Get PDF
    We study billiard dynamics inside an ellipse for which the axes lengths are changed periodically in time and an O(δ)O(\delta)-small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the billiard is no longer conserved. We show a Fermi acceleration in such system: there exists a billiard trajectory on which the energy tends to infinity. The construction is based on the analysis of dynamics in the phase space near a homoclinic intersection of the stable and unstable manifolds of the normally hyperbolic invariant cylinder Λ\Lambda, parameterised by the energy and time, that corresponds to the motion along the major axis of the ellipse. The proof depends on the reduction of the billiard map near the homoclinic channel to an iterated function system comprised by the shifts along two Hamiltonian flows defined on Λ\Lambda. The two flows approximate the so-called inner and scattering maps, which are basic tools that arise in the studies of the Arnold diffusion; the scattering maps defined by the projection along the strong stable and strong unstable foliations Wss,uuW^{ss,uu} of the stable and unstable invariant manifolds Ws,u(Λ)W^{s,u}(\Lambda) at the homoclinic points. Melnikov type calculations imply that the behaviour of the scattering map in this problem is quite unusual: it is only defined on a small subset of Λ\Lambda that shrinks, in the large energy limit, to a set of parallel lines t=constt=const as δ0\delta\to 0.Comment: 25 page

    Homoclinic Orbits In Slowly Varying Oscillators

    Get PDF
    We obtain existence and bifurcation theorems for homoclinic orbits in three-dimensional flows that are perturbations of families of planar Hamiltonian systems. The perturbations may or may not depend explicitly on time. We show how the results on periodic orbits of the preceding paper are related to the present homoclinic results, and apply them to a periodically forced Duffing equation with weak feedback

    Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups

    Get PDF
    Abstract not availabl
    corecore